Permutation Applets
Here you will find the collection of applets written by André
Heck and Leendert van Gastel for the Chapter on Permutations
in the interactive text on algebra. This package is part of the IDA project
at the University of Eindhoven. The package consists of
- a computational engine
- an applet showing different appearences
of a permutation
- a calculator applet that can be used
as a tool
- a game applet, that intuitionalizes the theorem
that any even permutation can be written as a product of 3-cycles.
The goal of the tabloid applet is to
show different appearences of a
permutation. The different appearences can be edited. A change
in one, leads to updated versions in the others. We have here four
different formats:
- list of cycles
- list of images of the numbers involved
- arrow view of the mapping (looks like mikado)
- the associated matrix
The following optional HTML parameters may be set:
- matrix
- "yes" means that the associated matrix is shown, "no" means that it is not shown
- arrows
- "yes" means that the arrow view is shown, "no" means that it is not shown
- bijection
- "yes" means that the list of images of the bijection is shown, "no" means that it is not shown
- cyclelist
- "yes" means that the list of cycles is shown, "no" means that it is not shown
- backgroundcolor
- "FFFFFF" means white, "000000" means black
Some remarks:
- Apart from changing the images, the permutation can be resized.
We implicitly assume that a permutation on n letters is also a
permutation on n+1 letters. So you can make a permutation also
smaller, if the permutation is trivial for the higher letters.
- In the cycles notation, it is allowed to give non-disjoint
cycles, like (1, 2, 3)(1, 2). As soon as is pressed, or any other
action is done, the permutation is rewritten as a series of disjoint
cycles.
- Sorting is quite natural in the mikado variant. As the actions
are all transpositions, the fact that sorting is possible, is equivalent to
the theorem that all permutations can be written as product of transpositions.
So we do not need a separate applet for that.
- In the matrix variant, moving columns is in fact the right
multiplication by cycles, moving rows the left-multiplication. This can
also be seen in the mikado variant. You may move the top of a line
which will correspond to right multiplication, or the bottom end
of a line, which will correspond to left multiplication.
- The sign of a permutation is visualised by the
number of crossings in the mikado variant. We take here as a definition
for the sign the number of displacements.
- The matrix appearence was initially made by Garth Dickie.
- The help facility yields a statement depending on the position
of the mouse.
Inspired by the calculators of Gord Simons, we developed a
Permutation Calculator that offers the
most common operations. Just push
The following optional HTML parameters may be set:
- maximumsize
- determining the maximum number of letters n
of Sn. The only restriction is due to
the capacity of TextField, which may differ among various platforms.
- backgroundcolor
- "FFFFFF" means white, "000000" means black
- fontsize
- "14" will give a 14-point fontsize
André Heck, Leendert van Gastel
Last modified: Mon Feb 24 10:32:37 MET